The monotonic sequence theorem and measurement of lengths and areas in axiomatic non-standard hyperrational analysis

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Аннотация

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

Язык оригиналаанглийский
Номер статьи42
ЖурналAxioms
Том8
Номер выпуска2
DOI
СостояниеОпубликовано - 1 июн 2019

Предметные области Scopus

  • Анализ
  • Алгебра и теория чисел
  • Математическая физика
  • Логика
  • Геометрия и топология

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